Categories, Color Gestures, and the "Souvenir Theorem": using Math to navigate the Complexity of Music and the Visual Arts

Arts, including music, seem to be far away from mathematical rigor. However, some mathematical constructions can well describe artistic forms and features and image/music translations. With the language of categories, in fact, we can describe forms as morphisms connecting points in space; form transformations, as natural transformations; visual-to-music form and structure mappings, as functors (mappings between categories). Effective translations verify gestural similarity: `similar' visual sketches and articulations/melodic profiles appear as being generated by the same gesture. We can define color gestures as mappings between points in the RGB space, and timbre gestures as mappings between timbres. Timbre-color correspondence requires equivalence classes and quotient categories. A timbre is mapped onto a color-class of colors and vice versa, satisfying a perceptive analogy between color and timbre bands (e.g., tension/relaxation feeling). We also present here the Souvenir Theorem (ST) and the Art Conjecture (AC). According to AC, the same idea can be identifiable and well-defined through its embodiments in similar artworks from different media. ST, with practical implications, is a condition of minimal recognizability of a form, allowing the creation of musical souvenirs.